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Ce travail porte sur des équations aux dérivées partielles issues de la physique mathématique, plus particulièrement sur celles régissant la supraconductivité. Ainsi, la majorité du travail concerne le modèle de Ginzburg-Landau, qui est un modèle macroscopique de supraconducteurs de type-II. Ce travail est divisé en deux parties principales: La première partie se focalise sur l'analyse des vortex du modèle de Ginzburg-Landau en deux dimensions pour les supraconducteurs de type-II, modèle conduisant à une estimation de la variation du nombre de vortex et à l'optimalité du réseau d'Abrikosov parmi les réseaux de Bravais. Nous avons également étudié certains modèles de stuctures des matériaux comme ceux de Lennard-Jones et de Thomas-Fermi. La seconde partie est consacrée à la fonctionnelle de Ginzburg-Landau en dimension $n$. Deux résultats principaux sont obtenus. L'un porte sur l'énergie renormalisée pour les minimiseurs de la fonctionnelle de Ginzburg-Landau. L'autre concerne les limites des solutions de l'équation de Ginzburg-Landau. Ces deux résultats sont fortement reliés aux applications $n$-harmoniques / Our work focus on the elliptic partial differential Equations arising from the mathematical physics, especially from the superconductivity. Therefore most of our work is on the Ginzburg-Landau model, which is a macroscopic model for the type-II superconductors. The work is divided into two big parts : this first part is on the vortices analysis of the Ginzburg-Landau model of Type-II superconductors in 2 dimensions, including the variations of the number of vortices and optimality of Abrikosov lattices among Bravais Lattices. We also have done some work related to the material structure, for example, the Lennard-Jones model and the Thomas-Fermi model. This second part is on the Ginzburg-Landau functional in $n$-dimensional case. Two main results are contained in this part: One is on the renormalized energy for minimizer of $n$-dimensional Ginzburg-Landau functional; The other one is on the limits of solutions to Ginzburg-Landau equations in $n$ dimension. Both of these two results are closely related to the p-harmonic maps

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-08-18T13:26:02Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao_FILLIPE_CESAR_07884743400.pdf: 6203092 bytes, checksum: 87063a6e43a5b0451757c513e7255e3e (MD5) / Made available in DSpace on 2016-08-18T13:26:02Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao_FILLIPE_CESAR_07884743400.pdf: 6203092 bytes, checksum: 87063a6e43a5b0451757c513e7255e3e (MD5) Previous issue date: 2015-07-31 / CNPq / O avan¸co tecnol´ogico tem permitido a constru¸c˜ao de sistemas supercondutores de baixas dimensionalidades, da ordem do comprimento de coerˆencia, ξ, e/ou comprimento de penetra¸c˜ao, λ. Esses sistemas s˜ao chamados de supercondutores mesosc ´opicos, onde se ´e capaz de alcan¸car correntes e campos cr´ıticos mais elevados. Dependendo do tamanho do supercondutor mesosc´opico, configura¸c˜oes de mol´eculas de v´ortices s˜ao obtidas tanto experimentalmente, como teoricamente usando a teoria de GL e a teoria de London. Nesse trabalho, utilizaremos o m´etodo variacional aplicado `a teoria de Ginzburg-Landau e a teoria de London para estudar v´ortices em discos supercondutores finos em campo magn´etico externo homogˆeneo e confinados em um potencial produzido por um dipolo magn´etico, respectivamente. Estudamos, inicialmente, discos supercondutores mesosc´opicos imersos em um meio isolante na presen¸ca de um campo magn´etico uniforme perpendicular ao plano do disco. Desprezamos os efeitos de desmagnetiza¸c˜ao e expandimos o parˆametro de ordem em fun¸c˜oes ortonormais do operador energia cin´etica. Para a obten¸c˜ao de v´ortices gigantes, representamos o parˆametro de ordem por apenas uma autofun¸c˜ao. Para estados de multiv´ortices, o parˆametro de ordem ´e representado como uma expans ˜A £o de duas autofun¸c˜oes, com vorticidades distintas. Observamos o comportamento da varia¸c˜ao da energia livre em fun¸c˜ao do campo magn´etico aplicado, para diversos tamanhos de discos supercondutores, acompanhando seus estados mais favor ´aveis. Na segunda parte da disserta¸c˜ao, estudamos a disposi¸c˜ao dos v´ortices e antiv´ortices confinados em um disco supercondutor atrav´es de campo magn´etico n˜ao homogˆeneo gerado por um dipolo magn´etico na dire¸c˜ao do eixo de simetria do disco, que possibilita a coexistˆencia de v´ortices e antiv´ortices, com diferentes configura¸c˜oes poss´ıveis. Para obtermos os estados com menor energia dentre as possibildades testadas, utilizamos como parˆametro de controle trˆes vari´aveis: a distˆancia radial ao eixo de simetria do dipolo, r0, a altura do dipolo em rela¸c˜ao ao disco, z0, e a magnitude do dipolo magn´etico, m0. Buscamos o valor da magnitude do dipolo no qual o estado sem v´ortices deixa de ser o metaest´avel, permitindo a existˆencia de estados com v´ortices. Os resultados mostram uma competi¸c˜ao entre o potencial de confinamento, efeitos produzidos pela borda do supercondutor e a intera¸c˜ao entre v´ortices, para manter os v´ortices e antiv´ortices no interior do disco. / Technological breakthroughs have allowed the construction of superconducting systems in low dimensionalities, which are of the order of the coherence length, ξ, and/or of the penetration depht, λ. These systems are called mesoscopic superconductors, which are capable of achieving high critical current and fields. Depending on the size of the mesoscopic superconducting, configurations of vortices molecules are obtained either experimentally or theoretically using the GL theory and the theory of London. In the present work, we apply the variational method to the Ginzburg- Landau theory and also the London theory to study vortices in thin superconductors disks submitted to an inhomogeneous magnetic field and confined to a potential produced by a magnetic dipole, respectively. We studied initially mesoscopic superconducting disks immersed in a insulation mean in the presence of a uniform magnetic field which is perpendicular to the plane of the disk. We neglect the demagnetization effects and expand the order parameter in orthonormal functions of the kinetic energy operator. To obtain giant vortices, the order parameter has only one eigenfunction. For multivortex states, the order parameter consists of two eigenfunctions, with different vorticities. We study the behaviour of free energy variation due to the applied magnetic field for different sizes of superconductor disks, and for different vortex states. In the second part of the dissertation, we study the arrangement of vortices and antivortices in a superconducting disk submitted to a non-homogeneous magnetic field generated by an out-plane magnetic dipole, which allows the coexistence of vortices and antivortices on different configurations. To get the states with lowest energy among the tested possibilities, we used three variables as control parameters: the radial distance from the disk symmetry axis to the dipole axis of symmetry, r0, the height of the dipole relative to the disc, z0, and the magnitude of the magnetic dipole, m0. We seek the value of the magnitude of the dipole in which the state with no vortices ceases to be the most stable one, allowing the existence of states with vortices. The results show a competition among the confinement potential, effects produced by the edge of the superconductor, and the interaction between vortices to keep the vortices and antivortices within the disc.

Made available in DSpace on 2018-03-22T15:55:39Z (GMT). No. of bitstreams: 1 tese_11668_Tese de doutorado - Fernando H. S. Eleutério.pdf: 1773499 bytes, checksum: e9c2c4b8631ab20a64b11ecf5e0aa507 (MD5) Previous issue date: 2017-12-15 / O foco deste trabalho é a investigação das duas fases intrínsecas de supercondutores estruturais, utilizando como objeto de estudo a resposta magnética e a sua composição granular. Ao investigar a susceptibilidade magnética ac na cerâmica policristalina a base de mercúrio dopada com rênio (Hg0.82Re0.18Ba2Ca2Cu3O8.16) (Hg,Re)- 1223) com o precursor com teor ótimo de oxigênio, pulverizada (reduzida a pó e peneirada) foi possível determinar a temperatura crítica da amostra e observar que ela depende do tamanho de cada do grão. Para a partícula de 20 μm foi encontrada somente uma temperatura crítica de 133K, porém são observadas duas temperaturas críticas (133K e 98 K) ao ser reduzir ainda mais o tamanho da partícula, a uma escala mesoscópica de 600 nm. Este fato foi elucidado por uma extensão da teoria de Ginzburg-Landau utilizando dois parâmetros de ordem para parametrizar o campo escalar complexo e fazendo a conexão com o modelo microscópico responsável por descrever as peculiaridades destes tipos de grãos.

Made available in DSpace on 2018-08-01T21:59:52Z (GMT). No. of bitstreams: 1 tese_11668_Tese final Fernando Henrique Santos Eleutério - PPGFis20180316-142248.pdf: 1478639 bytes, checksum: 8a421c51018e8816f936d7cac9314677 (MD5) Previous issue date: 2017-12-15 / O foco deste trabalho é a investigação das duas fases intrínsecas de supercondutores estruturais, utilizando como objeto de estudo a resposta magnética e a sua composição granular. Ao investigar a susceptibilidade magnética ac na cerâmica policristalina a base de mercúrio dopada com rênio (Hg0.82Re0.18Ba2Ca2Cu3O8.16) (Hg,Re)- 1223) com o precursor com teor ótimo de oxigênio, pulverizada (reduzida a pó e peneirada) foi possível determinar a temperatura crítica da amostra e observar que ela depende do tamanho de cada do grão. Para a partícula de 20 μm foi encontrada somente uma temperatura crítica de 133K, porém são observadas duas temperaturas críticas (133K e 98 K) ao ser reduzir ainda mais o tamanho da partícula, a uma escala mesoscópica de 600 nm. Este fato foi elucidado por uma extensão da teoria de Ginzburg-Landau utilizando dois parâmetros de ordem para parametrizar o campo escalar complexo e fazendo a conexão com o modelo microscópico responsável por descrever as peculiaridades destes tipos de grãos.

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Orientador: Rafael Zadorosny / Resumo: Este trabalho está estruturado em duas partes. A primeira, envolve o estudo do estado de vórtices e comportamento magnético de um supercondutor mesoscópico submetido a um gradiente térmico. Foi veri cado que a influência de um gradiente térmico altera o estado de vórtice gigante e pode levar a magnetização a ter um comportamento anômalo, o qual depende do parâmetro de Ginzburg-Landau e das temperaturas nas regiões "quente" e "fria". Na segunda parte, abordamos o estado resistivo durante o movimento de um vórtice e durante a dinâmica de par vórtice-antivórtice (V-AV) para os tipos de Abrikosov e Cinemáticos. Estudamos quais parâmetros afetam as propriedades do não equilíbrio e os mecanismos de dissipação presentes no processo de difusão térmica. Para os processos envolvendo vórtice ou V-AV de Abrikosov, os mecanismos de dissipação devido à processos de relaxação contribuem para a maior parte da potência total dissipada e, consequentemente, exercem uma grande contribuição para a variação de temperatura. Por outro lado, para o caso de V-AV cinemáticos, os elétrons normais aparecem como principal mecanismo de dissipação, todavia os processos de relaxação tem uma contribuição considerável para esse caso. Para realizar tais estudos, utilizamos o teorema da energia livre e as equações de Ginzburg- Landau generalizadas. / Abstract: This work is structured in two parts. In the rst one, the vortices states and the magnetic behavior were studied for a mesoscopic superconductor under a thermal gradient. It was veri ed that the thermal gradient changes the giant vortex state and could lead a exotic behavior of the magnetization curve, depending on the parameter and the temperature of the hot and the cold . In the second part, the focus was the resistive state during one vortex motion and during the annihilation process of a vortex-antivortex (V-AV) pair both with Abrikosov and Kinematic types. We de ned which parameters a ect the nonequilibrium proprieties and the dissipative mechanisms present in the thermal di usion processes. For the cases involving a vortex or an Abrikosov's V-AV pair, the dissipative mechanisms due to the relaxation process have the major contribution for the total power dissipated power and consequently, exert great contribuition on the temperature variation. On the other hand, for the case of kinematic V-AV, the normal eletrons rised as the main dissipative mechanism, however relaxation process has a considerable contribuition for this case. To realize such studies, we used the free energy theorem and the generalized Ginzburg-Landau equations. / Doutor

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a critical point $u^\ep\in H^1(N;\mathbb{C})$ of the Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^2}{2\ep^2}\ed and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$-converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$. Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on $\mathbb{R}^3$, we show the Ginzburg-Landau functional $E^\ep$ defined above $\Gamma$-converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a critical point $u^\ep\in H^1(N;\mathbb{C})$ of the Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^2}{2\ep^2}\ed and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$-converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$. Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on $\mathbb{R}^3$, we show the Ginzburg-Landau functional $E^\ep$ defined above $\Gamma$-converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

In part I, a numerical study of the mixed states in a mesoscopic type-II superconducting cylinder is described. Steady-state configurations and transient behavior of the magnetic vortices for various values of the applied magnetic field H are presented. Transitions between different multi-vortex states as H is changed is demonstrated by abrupt changes in vortex configurations and jumps in the B vs H plot. An efficient scheme to determine the equilibrium vortex configuration in a mesoscopic system at any given applied field, not limited to the symmetry of the system, is devised and demonstrated. In part II, a superconducting thin film is subject to a non-uniform magnetic field from a vertical magnetic dipole, consisting of two magnetic monopoles of opposite charges. For a film with constant thickness and with no pins, it has been found that the film carries two pairs of vortex-antivortex in the steady state in the imposed flux range of 2.15 < (Phi)+ < 2.90 (in units of flux quantum) and no vortex at all for (Phi)+ <= 2.15. Transitions from a superconducting state with 3 pairs of vortex-antivortex to one with 2 pairs, where a pair of vortex-antivortex annihilates, have been observed in the pseudo-time sequence. With a perturbation with antidots (holes), vortexantivortex pair has been created for lower magnetic fluxes down to (Phi)+ = 1.3. In the sample of size 16(Xi) x 16(Xi), the attraction force between the vortex and antivortex always dominates over the pinning force, so that they eventually come out of pins, move toward each other, and annihilate each other. The annihilation rate, measured with time taken for the annihilation, is reduced noticeably by the increase of the distance between pins, or the increase in the pin size. A simulation of the magnetic vortex pinning in the sample of size 32(Xi) x 32(Xi) suggests we are likely to achieve pinning of the vortex-antivortex pair with the sample size around this and vortex-antivortex separation of 22(Xi). Using this sample as a template, the maximum density of pinned vortices achievable is calculates to be about 7.6 x 10^14 vortices/m2 for (Xi) =~ 1.6A°.